The dichotomy for FO model checking in tournaments also coincides with a more combinatorial one
related to the growth function \(g_\mathcal {T}(n)\)  of the class \(\mathcal {T}\) ,
defined as the number of tournaments of \(\mathcal {T}\)  with \(n\)  vertices counted up to isomorphism.
We show that the growth of \(\mathcal {T}\)  is at most \(c^n\)  if and only if its twin-width is bounded.
To the contrary, when twin-width is not bounded, the growth jump to at least \((\left\lfloor n/2 \right\rfloor -1)!\) .
This exponential/factorial gap can be seen as a generalization of
the Marcus-Tardos theorem for permutations avoiding a fixed pattern [1]}.
